This same exact issue is happening to me as well! I will have good connection and when I'm in champion select PvP it will require me to click and re-click to either confirm my character or to Ban one. Same issue is reoccurring and was an issue for other users in my game not just me.
This happened twice on Saturday around noon central time, once on Friday around 2-4, and last weekend Saturday in the morning.
Also on Saturday in a bot game I loaded after the game started when everyone else was at level 2 / 3. I barely got to pick a player then a loading screen came on saying I was not connected but if I tried to disconnect I would be penalized.
I realize that this could be an issue with my end as well but there is no way for me to accurately check while in game so how can I do anything about this!?
I did manage some screenshots of both issues though they don't seem to be much help.